the 7-adic expansion of square root of 305:= 2.2222222(index7) this is one representation of root of 305 but there is another representation of 7-adic expansion

Hence they have the same 7-adic expansion . therefore the distance is 0 in Q7?

Is my attemption right? ( i am not sure whether i caculated these two 7-adic expansion in a right way)

Nov 19th 2013, 04:27 PM

SlipEternal

Re: 7-adic distance in Q7

So, you are looking for such that . So, suppose (it can be 2 or 5). Then , so . That gives , and a carry of 1. So, . This gives as you suggested. Again, the carry is 1. Then, , so and the carry is 3. Next, . So, , and the carry is 5. So, I don't know how you calculated the square root of 305, but I got a very different result.

Nov 19th 2013, 05:20 PM

Sonifa

Re: 7-adic distance in Q7

yes ur right, i got some mistakes of my caculation. now, i got the new representation for root 305 should be right: 2.22345... or
5.4432....

and my question is how to caculate the distance between -1/3
and root of 305 in Q7 (i.e. w.r.t the norm|.|p) Hints for the question is:
compute first few terms of 7-adic expansion of -1/3 and root 305 .

Now , i dont know how to consider the distance..

Nov 19th 2013, 05:47 PM

SlipEternal

Re: 7-adic distance in Q7

The easiest way to check if a solution is correct is to use evaluate its geometric sum: . So, you found the correct representation for . So, is the minimum distance between the possible square roots of . So, if and , and both of them squared give , the 7-adic distance is

Nov 19th 2013, 05:58 PM

Sonifa

Re: 7-adic distance in Q7

sorry, do you mean if i continue to do the method for more digits of 2.22345... or 5.4432.... then i can get k and -k as ur post?? but how could u ensure that for 2.22345... , u can only get 5 on the digits after 4？？？

Nov 19th 2013, 07:00 PM

SlipEternal

Re: 7-adic distance in Q7

You are writing . That is not a notation I am familiar with. I am used to which means . I suspect they mean the same thing. I am using only digits to the left of the decimal place. You are using digits to the right of the decimal place. I learned that . Anyway, using your notation, you want . Note that .

Nov 20th 2013, 12:25 AM

Sonifa

Re: 7-adic distance in Q7

Thats clear for me now. thanks a lot

Nov 21st 2013, 01:50 AM

lahuxixi

Re: 7-adic distance in Q7

How did you calculate the 7-adic expansion of square root 305? i honestly cannot do it

Nov 21st 2013, 03:17 AM

ineedhelplz

Re: 7-adic distance in Q7

Quote:

Originally Posted by lahuxixi

How did you calculate the 7-adic expansion of square root 305? i honestly cannot do it

me too

Nov 21st 2013, 04:41 AM

SlipEternal

Re: 7-adic distance in Q7

Calculate it digit-by-digit. I went through it in post #2.

Nov 21st 2013, 04:59 AM

lahuxixi

Re: 7-adic distance in Q7

I don't need the answer to the question of this thread. I want to ask you about the method of calculating the expansion for other cases. Thank you